The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910701435486208 |
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| author | Hall, Lucas Huang, Leonard Krajczok, Jacek Tobolski, Mariusz |
| author_facet | Hall, Lucas Huang, Leonard Krajczok, Jacek Tobolski, Mariusz |
| contents | The Stone-von Neumann Theorem is a fundamental result which unified the competing quantum mechanical models of matrix mechanics and wave mechanics. It's mechanism of proof ultimately involved the study of unitary group representations on a Hilbert space. In this article, we continue the broad generalization set out in arxiv:1903.09351 and arxiv:2109.08997, analyzing representations of locally compact quantum dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models which appear in the literature, and for certain coactions (G, A, α) recover the multiplicity results of arxiv:2109.08997. As a corollary, we develop a new criterion for identifying strongly regular locally compact quantum groups, related to the study of their dynamics on elementary C*-algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_15264 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups Hall, Lucas Huang, Leonard Krajczok, Jacek Tobolski, Mariusz Operator Algebras 46L55 (Primary), 47L55 (Secondary) The Stone-von Neumann Theorem is a fundamental result which unified the competing quantum mechanical models of matrix mechanics and wave mechanics. It's mechanism of proof ultimately involved the study of unitary group representations on a Hilbert space. In this article, we continue the broad generalization set out in arxiv:1903.09351 and arxiv:2109.08997, analyzing representations of locally compact quantum dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models which appear in the literature, and for certain coactions (G, A, α) recover the multiplicity results of arxiv:2109.08997. As a corollary, we develop a new criterion for identifying strongly regular locally compact quantum groups, related to the study of their dynamics on elementary C*-algebras. |
| title | The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups |
| topic | Operator Algebras 46L55 (Primary), 47L55 (Secondary) |
| url | https://arxiv.org/abs/2312.15264 |